By performing DFPT calculations on SrTiO\(_3\)  we evaluated the FES mode energy for different functionals and found that for the respective optimized lattice parameters (indicated by the vertical dashed lines), the FES mode is unstable for PBE and PBEsol but not for LDA, see Fig. 1(e) and Tab. 1.
This indicates that LDA would predict a classical paraelectric ground state in contrast with PBE and PBEsol (as well as hybrid functionals [1]}, [2]}) predicting a phase transition.
Regardless of the functional an additional frequency softening of the FES mode is induced by the \(c\)  lattice parameter elongation.
This instability can be understood by considering the potential energy landscape of the FES mode computed with PBE.
Given the strong dependence of the DFPT results on the lattice parameter \(c\)  we calculated the potential energy surface for different \(c\)  values, denoted by \(Q_{\rm {c}}\) , as reported in Fig. 1(f).
The potential energy surface for the FES mode is evaluated by displacing the atomic positions along the optimized ferroelectric geometry with respect to the optimized tetragonal geometry; we denote this parameterized displacement as \(Q_{\rm {f}}\) .
At the optimized lattice parameter, \(Q_{\rm {c}}=0\)  Å, the potential energy surface shows a shallow double well potential.
This explains the instability found in DFPT and why in absence of quantum fluctuations the system would spontaneously collapse into one of the two wells inducing a ferroelectric polarisation.
Similar potential energy calculations for the LDA functional (not shown) provide a single well dispersion and hence no phase transition is expected.
Changing \(Q_{\rm {c}}\)  results in an asymmetric behaviour when using the PBE functional: for negative \(Q_{\rm {c}}\) , i.e. lattice contraction, the double well disappears while it deepens under lattice expansion, positive \(Q_{\rm {c}}\) .
This behavior explains why the phase transition between the ferroelectric and paraelectric phases of SrTiO\(_3\)  can be easily induced by strain [3]}, [4]}.
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